Integral expressions for the numerical evaluation of product form expressions over irregular multidimensional integer state spaces

Abstract

We consider stochastic systems defined over irregular, multidimensional, integer spaces that have a product form steady state distribution. Examples of such systems include closed and BCMP type of queuing networks, polymerization and genetic models. In these models the system state is a vector of integers, n=[n 1,...,n M ] and the steady state solution has product form of the type ?(n)=? i=1 M f i (n i ). To obtain useful statistics from such product form solutions, ?(n) has to be summed over some subset of the space over which it is defined. We consider situations when these subsets are defined by a set of equalities and inequalities with integer coefficients, as is most often the case and provide integral expressions to obtain these sums. Typically, a brute force technique to obtain the sum is computationally very expensive. Algorithmic solutions are available for only specific forms of f i (n i ) and shapes of the state space. In this paper we derive general integral expressions for arbitrary state spaces and arbitrary f i (n i ). The expressions that we derive here become especially useful if the generating functions f i (n i ) can be expressed as a ratio of polynomials in which case, exact closed form expressions can be obtained for the sums. We demonstrate the wide applicability of the integral expressions that we derive here through three examples in which we model finite highway cellular systems, copy networks in multicast packet switches and a BCMP queuing network modeling a multiuser computer system.